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Topological recursion

From Wikipedia, the free encyclopedia

In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory.

Introduction

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The topological recursion is a construction in algebraic geometry.[1] It takes as initial data a spectral curve: the data of , where: is a covering of Riemann surfaces with ramification points; is a meromorphic differential 1-form on , regular at the ramification points; is a symmetric meromorphic bilinear differential form on having a double pole on the diagonal and no residue.

The topological recursion is then a recursive definition of infinite sequences of symmetric meromorphic n-forms on , with poles at ramification points only, for integers g≥0 such that 2g-2+n>0. The definition is a recursion on the integer 2g-2+n.

In many applications, the n-form is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2-2g+n the Euler characteristics, whence the name "topological recursion".

Schematic illustration of the topological recursion: recursively adding pairs of pants to build a surface of genus g with n boundaries

Origin

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The topological recursion was first discovered in random matrices. One main goal of random matrix theory, is to find the large size asymptotic expansion of n-point correlation functions, and in some suitable cases, the asymptotic expansion takes the form of a power series. The n-form is then the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found[2][3][4] that the coefficients always obey a same recursion on 2g-2+n. The idea to consider this universal recursion relation beyond random matrix theory, and to promote it as a definition of algebraic curves invariants, occurred in Eynard-Orantin 2007[1] who studied the main properties of those invariants.

An important application of topological recursion was to Gromov–Witten invariants. Marino and BKMP[5] conjectured that Gromov–Witten invariants of a toric Calabi–Yau 3-fold are the TR invariants of a spectral curve that is the mirror of .

Since then, topological recursion has generated a lot of activity in particular in enumerative geometry. The link to Givental formalism and Frobenius manifolds has been established.[6]

Definition

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(Case of simple branch points. For higher order branchpoints, see the section Higher order ramifications below)

  • For and :

where is called the recursion kernel:
and is the local Galois involution near a branch point , it is such that . The primed sum means excluding the two terms and .

  • For and :



with any antiderivative of .

  • The definition of and is more involved and can be found in the original article of Eynard-Orantin.[1]

Main properties

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  • Symmetry: each is a symmetric -form on .
  • poles: each is meromorphic, it has poles only at branchpoints, with vanishing residues.
  • Homogeneity: is homogeneous of degree . Under the change , we have .
  • Dilaton equation:


where .

  • Loop equations: The following forms have no poles at branchpoints


where the sum has no prime, i.e. no term excluded.

  • Deformations: The satisfy deformation equations
  • Limits: given a family of spectral curves , whose limit as is a singular curve, resolved by rescaling by a power of , then .
  • Symplectic invariance: In the case where is a compact algebraic curve with a marking of a symplectic basis of cycles, is meromorphic and is meromorphic and is the fundamental second kind differential normalized on the marking, then the spectral curve and , have the same shifted by some terms.
  • Modular properties: In the case where is a compact algebraic curve with a marking of a symplectic basis of cycles, and is the fundamental second kind differential normalized on the marking, then the invariants are quasi-modular forms under the modular group of marking changes. The invariants satisfy BCOV equations.[clarification needed]

Generalizations

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Higher order ramifications

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In case the branchpoints are not simple, the definition is amended as follows[7] (simple branchpoints correspond to k=2):

The first sum is over partitions of with non empty parts , and in the second sum, the prime means excluding all terms such that .

is called the recursion kernel:

The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants will not depend on it.

Topological recursion invariants and intersection numbers

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The invariants can be written in terms of intersection numbers of tautological classes

[8]

(*)
where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus , and smooth labeled marked points , and equipped with a map . is the Chern class of the cotangent line bundle whose fiber is the cotangent plane at . is the th Mumford's kappa class. The coefficients , , , are the Taylor expansion coefficients of and in the vicinity of branchpoints as follows: in the vicinity of a branchpoint (assumed simple), a local coordinate is . The Taylor expansion of near branchpoints , defines the coefficients
.
The Taylor expansion at , defines the 1-forms coefficients
whose Taylor expansion near a branchpoint is
.
Write also the Taylor expansion of
.
Equivalently, the coefficients can be found from expansion coefficients of the Laplace transform, and the coefficients are the expansion coefficients of the log of the Laplace transform
.

For example, we have

The formula (*) generalizes ELSV formula as well as Mumford's formula and Mariño-Vafa formula.

Some applications in enumerative geometry

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Mirzakhani's recursion

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M. Mirzakhani's recursion for hyperbolic volumes of moduli spaces is an instance of topological recursion. For the choice of spectral curve
the n-form is the Laplace transform of the Weil-Petersson volume

where is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths , and is the Weil-Petersson volume form.
The topological recursion for the n-forms , is then equivalent to Mirzakhani's recursion.

Witten–Kontsevich intersection numbers

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For the choice of spectral curve
the n-form is

where is the Witten-Kontsevich intersection number of Chern classes of cotangent line bundles in the compactified moduli space of Riemann surfaces of genus g with n smooth marked points.

Hurwitz numbers

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For the choice of spectral curve
the n-form is

where is the connected simple Hurwitz number of genus g with ramification : the number of branch covers of the Riemann sphere by a genus g connected surface, with 2g-2+n simple ramification points, and one point with ramification profile given by the partition .

Gromov–Witten numbers and the BKMP conjecture

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Let a toric Calabi–Yau 3-fold, with Kähler moduli . Its mirror manifold is singular over a complex plane curve given by a polynomial equation , whose coefficients are functions of the Kähler moduli. For the choice of spectral curve with the fundamental second kind differential on ,
According to the BKMP[5] conjecture, the n-form is

where
is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into , with n boundaries mapped to a special Lagrangian submanifold . is the 2nd relative homology class of the surface's image, and are homology classes (winding number) of the boundary images.
The BKMP[5] conjecture has since then been proven.

Notes

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  1. ^ a b c Invariants of algebraic curves and topological expansion, B. Eynard, N. Orantin, math-ph/0702045, ccsd-hal-00130963, Communications in Number Theory and Physics, Vol 1, Number 2, p347-452.
  2. ^ B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, JHEP/024A/0904, hep-th/0407261 A short overview of the ”Topological recursion”, math-ph/arXiv:1412.3286
  3. ^ A. Alexandrov, A. Mironov, A. Morozov, Solving Virasoro Constraints in Matrix Models, Fortsch.Phys.53:512-521,2005, arXiv:hep-th/0412205
  4. ^ L. Chekhov, B. Eynard, N. Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612 (2006) 053, math-ph/0603003
  5. ^ a b c Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178,2009
  6. ^ P. Dunin-Barkowski, N. Orantin, S. Shadrin, L. Spitz, "Identification of the Givental formula with the spectral curve topological recursion procedure", Commun.Math.Phys. 328 (2014) 669-700.
  7. ^ V. Bouchard, B. Eynard, "Think globally, compute locally", JHEP02(2013)143.
  8. ^ B. Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves, math-ph: arxiv.1110.2949, Journal Communications in Number Theory and Physics, Volume 8, Number 3.

References

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[1]

  1. ^ O. Dumitrescu and M. Mulase, Lectures on the topological recursion for Higgs bindles and quantum curves, https://www.math.ucdavis.edu/~mulase/texfiles/OMLectures.pdf